Abstract
In this paper, we consider a differentiable multiobjective optimization problem with generalized cone constraints (for short, MOP). We investigate the relationship between weakly efficient solutions for (MOP) and for the multiobjective optimization problem with the modified objective function and cone constraints [for short, (MOP) η (x)] and saddle points for the Lagrange function of (MOP) η (x) involving cone invex functions under some suitable assumptions. We also prove the existence of weakly efficient solutions for (MOP) and saddle points for Lagrange function of (MOP) η (x) by using the Karush-Kuhn-Tucker type optimality conditions under generalized convexity functions. As an application, we investigate a multiobjective fractional programming problem by using the modified objective function method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aghezzaf B., Hachimi M.: Generalized invexity and duality in multiobjective programming problems. J. Global Optim. 18, 91–101 (2000)
Antczak T.: A new approach to multiobjective programming with a modified objective function. J. Global Optim. 27, 485–495 (2003)
Antczak T.: An η-approximation approach for nonlinear mathematical programming problems involving invex functions. Num. Funct. Anal. Optim. 25, 423–438 (2004)
Antczak T.: Modified ratio objective approach in mathematical programming. J. Optim. Theory Appl. 126, 23–40 (2005)
Antczak T.: An η-approximation method in nonlinear vector optimization. Nonlinear Anal. 63, 236–255 (2005)
Antczak T.: An η-approximation approach to duality in mathematical programming problems involving r-invex functions. J. Math. Anal. Appl. 315, 555–567 (2006)
Bonnans J.F., Shapiro A.: Perturbation Analysis of Optimization Problem. Springer-Verlag, New York (2000)
Craven, B.D.: Control and Optimization. Chapman & Hall (1995)
Crouzeix J.P., Martinez-Legaz, J.E., Volle, M.: Generalized convexity, generalized monotonicity, proceedings of the fifth symposium on generalized convexity, Kluwer Academic Publishers, Luminy, France, (1997)
Dutta J.: On generalized preinvex functions. Asia-Pacific J. Oper. Res. 18, 257–272 (2001)
Eguda R.R., Hanson M.A.: Multiobjective duality with invexity. J. Math. Anal. Appl. 126, 469–477 (1987)
Giannessi, F. (eds): Vector Variational Inequalities and Vector Equilibrium. Kluwer Academic Publishers, Dordrechet, Boston, London (2000)
Hanson M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Khan Z.A., Hanson M.A.: On ratio invexity in mathematical programming. J. Math. Anal. Appl. 205, 330–336 (1997)
Li L., Li J.: Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints. Appl. Math. Lett. 21, 599–606 (2008)
Luc D.T.: Theory of Vector Optimization. Springer-Verlag, Berlin (1989)
Mishra S.K., Wang S.Y., Lai K.K.: Optimality and duality in nondifferentiable and multiobjective programming under generalized d-invexity. J. Global Optim. 29, 425–438 (2004)
Weir T., Jeyakumar V.: A class of nonconvex functions and mathematical programming. Bull. Austral. Math, Soc. 38, 177–189 (1988)
Weir T., Mond B., Craven B.D.: On duality for weakly minimized vector valued optimization. Optimiz. 17, 711–721 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, J.W., Cho, Y.J., Kim, J.K. et al. Multiobjective optimization problems with modified objective functions and cone constraints and applications. J Glob Optim 49, 137–147 (2011). https://doi.org/10.1007/s10898-010-9539-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-010-9539-3
Keywords
- Multiobjective optimization problem
- Q-(pseudo)invex
- Q-convexlike
- Weakly efficient solution
- Saddlepoint
- KKT condition
- Weak (strong, converse) duality
- Lagrange function